An Extensible Natural Calculus for Argument Presentation

Abstract

This paper proposes the notion of an extensible natural calculus, within a general setting of a computational model for human deductive reasoning. This research is motivated by a problem almost any contemporary automated reasoning system has to confront, namely, that these systems generate proofs of increasing complexity and length, that are almost incomprehensible for a human user. Hence the necessity to transform these machine generated proofs from their internal format into a more abstract and more human oriented style. Researchers have long been aware of the need for an intermediate representation, to spilt the entire transformation process from the machine oriented representation via this intermediate representation finally into natural language. By analyzing the blueprint of the entire transformation process we first point out that this intermediate representation should have the quality of being the direct product of a human deductive apparatus or, at least an appropriate mathematical abstraction thereof. Then, based on our preliminary empirical study, we point out the inadequateness of using proofs based on Gentzen’s natural deduction formalism as such an intermediate representation and set out our own proposal, called detailed natural proofs (DNPs). Here the notion of a proof step is generalized, by associating it to the notion of primitive cognitive procedures, rather than restricting it solely to inference rules of natural logic or mental logic. The main part of this paper is concerned with a formalism in which the DNPs can be encoded, called an extensible natural calculus. This discussion is carried out within the general setting of a computational model for human formal deductive reasoning.